Low-Regularity Solutions of the Nonlinear Schr\"odinger Equation on the Spatial Quarter-Plane

Abstract

The Hadamard well-posedness of the nonlinear Schr\"odinger equation with power nonlinearity formulated on the spatial quarter-plane is established in a low-regularity setting with Sobolev initial data and Dirichlet boundary data in appropriate Bourgain-type spaces. As both of the spatial variables are restricted to the half-line, a different approach is needed than the one previously used for the well-posedness of other initial-boundary value problems. In particular, now the solution of the forced linear initial-boundary problem is estimated \textit{directly}, both in Sobolev spaces and in Strichartz-type spaces, i.e. without a linear decomposition that would require estimates for the associated homogeneous and nonhomogeneous initial value problems. In the process of deriving the linear estimates, the function spaces for the boundary data are identified as the intersections of certain modified Bourgain-type spaces that involve spatial half-line Fourier transforms instead of the usual whole-line Fourier transform found in the definition of the standard Bourgain space associated with the one-dimensional initial value problem. The fact that the quarter-plane has a corner at the origin poses an additional challenge, as it requires one to expand the validity of certain Sobolev extension results to the case of a domain with a non-smooth (Lipschitz) and non-compact boundary.